Presented at:
To appear in:
NIPS*10, Vancouver BC Canada, 6-9 December 2010.
http://www.nips.cc
Advances in Neural Information Processing Systems,
eds. J. Lafferty, C. Williams, R. Zemel, J. Shawe-Taylor, and A. Culotta
volume 23, May 2011, MIT Press, Cambridge MA.
Implicit encoding of prior probabilities
in optimal neural populations
Deep Ganguli and Eero P. Simoncelli
Howard Hughes Medical Institute, and
Center for Neural Science
New York University
New York, NY 10003
{dganguli,eero}@cns.nyu.edu
Optimal coding provides a guiding principle for understanding the representation
of sensory variables in neural populations. Here we consider the influence of a
prior probability distribution over sensory variables on the optimal allocation of
neurons and spikes in a population. We model the spikes of each cell as samples
from an independent Poisson process with rate governed by an associated tuning
curve. For this response model, we approximate the Fisher information in terms
of the density and amplitude of the tuning curves, under the assumption that tuning width varies inversely with cell density. We consider a family of objective
functions based on the expected value, over the sensory prior, of a functional of
the Fisher information. This family includes lower bounds on mutual information
and perceptual discriminability as special cases. In all cases, we find a closed
form expression for the optimum, in which the density and gain of the cells in
the population are power law functions of the stimulus prior. This also implies
a power law relationship between the prior and perceptual discriminability. We
show preliminary evidence that the theory successfully predicts the relationship
between empirically measured stimulus priors, physiologically measured neural
response properties (cell density, tuning widths, and firing rates), and psychophysically measured discrimination thresholds.
1
Introduction
Many bottom up theories of neural encoding posit that sensory systems are optimized to represent sensory information, subject to limitations of noise and resources (e.g., number of neurons,
metabolic cost, wiring length). It is difficult to test this concept because optimization of any formulation that attempts to correctly incorporate all of the relevant ingredients is generally intractable. A
substantial literature has considered population models in which each neuron’s mean response to a
scalar variable is characterized by a tuning curve [e.g., 1–6]. For these simplified models, several
papers have examined the optimization of Fisher information, as a bound on mean squared error
[7–10]. In these results, the distribution of sensory variables is assumed to be uniform and the populations are assumed to be homogeneous with regard to tuning curve shape, spacing, and amplitude.
The distribution of sensory variables encountered in the environment is often non-uniform, and it is
thus of interest to understand how variations in probability affect the design of optimal populations.
It would seem natural that a neural system should devote more resources to regions of sensory space
that occur with higher probability, analogous to results in coding theory [11]. At the single neuron
level, several publications describe solutions in which monotonic neural response functions allocate
greater dynamic range to higher probability stimuli [12–15]. At the population level, non-uniform
allocations of neurons with identical tuning curves have been shown to be optimal for non-uniform
stimulus distributions [16, 17].
Here, we examine the influence of a sensory prior on the optimal allocation of neurons and spikes
in a population, and the implications of this optimal allocation for subsequent perception. Given
a prior distribution over a scalar stimulus parameter, and a resource budget of N neurons with an
average of R spikes/sec for the entire population, we seek the optimal shapes, positions, and amplitudes of tuning curves. We assume a population with independent Poisson spiking, and consider
a family of objective functions based on Fisher information. We then approximate the Fisher information in terms of two continuous resource variables, the density and gain of the tuning curves.
This approximation allows us to obtain a closed form solution for the optimal population. For all
objective functions, we find that the optimal tuning curve properties (cell density, tuning width, and
gain) are power-law functions of the stimulus prior, with exponents dependent on the specific choice
of objective function. Through the Fisher information, we also derive a bound on perceptual discriminability, again in the form a power-law of the stimulus prior. Thus, our framework provides
direct and experimentally testable links between sensory priors, properties of the neural representation, and perceptual discriminability. We provide preliminary evidence that these relationships are
supported by experimental data.
2
Encoding model
We assume a conventional model for a population of N neurons responding to a single scalar variable, s [1–6]. The number of spikes emitted (per unit time) by the nth neuron is a sample from
an independent Poisson process, with mean rate determined by its tuning function, hn (s). The
probability density of the population response can be written as
p(r|s) =
N
hn (s)rn e−hn (s)
.
rn !
n=1
We also assume the total expected spike rate, R, of the population is fixed, which places a constraint
on the tuning curves:
p(s)
N
hn (s) ds = R,
(1)
n=1
where p(s) is the probability distribution of stimuli in the environment. We refer to this as a sensory
prior, in anticipation of its future use in Bayesian decoding of the population response.
3
Objective function
We now ask: what is the best way to represent values drawn from p(s) given the limited resources
of N neurons and R total spikes? To formulate a family of objective functions which depend on
both p(s), and the tuning curves, we first rely on Fisher information, If (s), which can be written as
a function of the tuning curves [1, 18]:
∂2
If (s) = − p(r|s) 2 log p(r|s) dr
∂s
N
2
hn (s)
=
.
h (s)
n=1 n
The Fisher information can be used to express lower bounds on mutual information [16], the variance of an unbiased estimator [18], and perceptual discriminability [19]. Specifically, the mutual
information, I(r; s), is bounded by:
1
2πe
I(r; s) ≥ H(s) −
p(s) log
ds,
(2)
2
If (s)
where H(s) is the entropy, or amount of information inherent in p(s), which is independent of the
neural population. The Cramer-Rao inequality allows us to express the minimum expected squared
2
stimulus discriminability achievable by any decoder1 :
p(s)
ds.
δ 2 ≥ ∆2
If (s)
(3)
The constant ∆ determines the performance level at threshold in a discrimination task.
We formulate a generalized objective function that includes the Fisher bounds on information and
discriminability as special cases:
N
N
h2 (s)
n
arg max
hn (s) ds = R,
(4)
p(s) f
ds,
s.t.
p(s)
h (s)
hn (s)
n=1 n
n=1
where f (·) is either the natural logarithm, or a power function. When f (x) = log(x), optimizing
Eq. (4) is equivalent to maximizing the lower bound on mutual information given in Eq. (2). We
refer to this as the infomax objective function. Otherwise, we assume f (x) = xα , for some exponent
α. Optimizing Eq. (4) with α = −1 is equivalent to minimizing the squared discriminability bound
expressed in Eq. (3). We refer to this as the discrimax objective function.
4
How to optimize?
The objective function expressed in Eq. (4) is difficult to optimize because it is non-convex. To
make the problem tractable, we first introduce a parametrization of the population in terms of cell
density and gain. The cell density controls both the spacing and width of the tuning curves, and the
gain controls their maximum average firing rates. Second, we show that Fisher information can be
closely approximated as a continuous function of density and gain. Finally, re-writing the objective
function and constraints in these terms allows us to obtain closed-form solutions for the optimal
tuning curves.
4.1
Density and gain for a homogeneous population
If p(s) is uniform, then by symmetry, the Fisher information for an optimal neural population should
also be uniform. We assume a convolutional population of tuning curves, evenly spaced on the unit
lattice, such that they approximately “tile” the space:
N
h(s − n) ≈ 1.
n=1
We also assume that this population has an approximately constant Fisher information:
N
h2 (s − n)
If (s) =
h(s − n)
n=1
=
N
φ(s − n) ≈ Iconv .
(5)
n=1
That is, we assume that the Fisher information curves for the individual neurons, φ(s − n), also
tile the stimulus space. The value of the constant, Iconv , is dependent on the details of the tuning
curve shape, h(s), which we leave unspecified. As an example, Fig. 1(a-b) shows that the Fisher
information for a convolutional population of Gaussian tuning curves, with appropriate width, is
approximately constant.
Now we introduce two scalar values, a gain (g), and a density (d), that affect the convolutional
population as follows:
n hn (s) = g h d(s − ) .
(6)
d
1
The conventional Cramer-Rao bound expresses the minimum mean squared error of any estimator, and in
general requires a correction for the estimator bias [18]. Here, we use it to bound the squared discriminability
of the estimator, as expressed in the stimulus space, which is independent of bias [19].
3
(b)
(a)
(c)
7
actual
approximate
8
6
5
4
3
2
1
(d)
(e)
Firing rate
Firing rate
approximate
Fisher
information
Fisher
information
actual
Fig. 1. Construction of a heterogeneous population of neurons. (a) Homogeneous population with
Gaussian tuning curves on the unit lattice. The tuning width of σ = 0.55 is chosen so that the curves
approximately tile the stimulus space. (b) The Fisher information of the convolutional population
(green) is approximately constant. (c) Inset shows d(s), the tuning curve density. The cumulative
integral of this density, D(s), alters the positions and widths of the tuning curves in the convolutional
population. (d) The warped population, with tuning curve peaks (aligned with tick marks, at locations
sn = D−1 (n)), is scaled by the gain function, g(s) (blue). A single tuning curve is highlighted
(red) to illustrate the effect of the warping and scaling operations. (e) The Fisher information of the
inhomogeneous population is approximately proportional to d2 (s)g(s).
The gain modulates the maximum average firing rate of each neuron in the population. The density
controls both the spacing and width of the tuning curves: as the density increases, the tuning curves
become narrower, and are spaced closer together so as to maintain their tiling of stimulus space. The
effect of these two parameters on Fisher information is:
N (d)
If (s) = d2 g
φ(ds − n)
n=1
≈ d2 g Iconv .
The second line follows from the assumption of Eq. (5), that the Fisher information of the convolutional population is approximately constant with respect to s.
The total resources, N and R, naturally constrain d and g, respectively. If the original (unit-spacing)
convolutional population is supported on the interval (0, Q) of the stimulus space, then the number
of neurons in the modulated population must be N (d) = Qd to cover the same interval. Under
the assumption that the tuning curves tile the stimulus space, Eq. (1) implies that R = g for the
modulated population.
4.2
Density and gain for a heterogeneous population
Intuitively, if p(s) is non-uniform, the optimal Fisher information should also be non-uniform. This
can be achieved through inhomogeneities in either the tuning curve density or gain. We thus generalize density and gain to be continuous functions of the stimulus, d(s) and g(s), that warp and scale
the convolutional population:
hn (s) = g(sn ) h(D(s) − n).
4
(7)
Infomax
Optimized function:
Density (Tuning width)−1
Gain
Fisher information
d(s)
g(s)
If (s)
Discriminability bound δmin (s)
f (x) = log x
Discrimax
−1
f (x) = −x
1
N p(s)
N p 2 (s)
− 12
R
Rp
∝ RN 2 p2 (s)
∝ p−1 (s)
(s)
1
2
∝ RN 2 p (s)
1
∝ p− 4 (s)
General
f (x) = −xα , α <
α−1
3α−1
(s)
2α
1−3α
(s)
Np
Rp
1
3
2
∝ RN 2 p 1−3α (s)
1
∝ p 3α−1 (s)
Table 1. Optimal heterogeneous population properties, for objective functions specified by Eq. (9).
s
Here, D(s) = −∞ d(t)dt, the cumulative integral of d(s), warps the shape of the prototype tuning
curve. The value sn = D−1 (n) represents the preferred stimulus value of the (warped) nth tuning
curve (Fig. 1(b-d)). Note that the warped population retains the tiling properties of the original
convolutional population. As in the uniform case, the density controls both the spacing and width
of the tuning curves. This can be seen by rewriting Eq. (7) as a first-order Taylor expansion of D(s)
around sn :
hn (s) ≈ g(sn ) h(d(sn )(s − sn )),
which is a generalization of Eq. (6).
We can now write the Fisher information of the heterogeneous population of neurons in Eq. (7) as
If (s) =
N
d2 (s) g(sn ) φ(D(s) − n)
n=1
2
≈ d (s) g(s) Iconv .
(8)
In addition to assuming that the Fisher information is approximately constant (Eq. (5)), we have
also assumed that g(s) is smooth relative to the width of φ(D(s) − n) for all n, so that we can
approximate g(sn ) as g(s) and remove it from the sum. The end result is an approximation of
Fisher information in terms of the continuous parameterization of cell density and gain. As earlier,
the constant Iconv is determined by the precise shape of the tuning curves.
As in the homogeneous case, the global resource values N and R will place constraints on d(s)
and g(s), respectively. In particular, we require
that D(·) map the entire input space onto the range
[1, N ], and thus D(∞) = N , or equivalently, d(s) ds = N . To attain the proper rate, we use the
fact that the warped tuning curves
sum to unity (before multiplication by the gain function) and use
Eq. (1) to obtain the constraint p(s)g(s) ds = R.
4.3
Objective function and solution for a heterogeneous population
Approximating Fisher information as proportional to squared density and gain allows us to re-write
the objective function and resource constraints of Eq. (4) as
p(s) f d2 (s) g(s) ds,
arg max
s.t.
d(s) ds = N, and
p(s)g(s) ds = R. (9)
d(s),g(s)
A closed-form optimum of this objective function is easily determined by taking the gradient of the
Lagrangian, setting to zero, and solving the resulting system of equations. Solutions are provided in
Table 1 for the infomax, discrimax, and general power cases.
In all cases, the solution specifies a power-law relationship between the prior, and the density and
gain of the tuning curves. In general, all solutions allocate more neurons, with correspondingly
narrower tuning curves, to higher-probability stimuli. In particular, the infomax solution allocates
an approximately equal amount of probability mass to each neuron. The shape of the optimal gain
function depends on the objective function: for α < 0, neurons with lower firing rates are used
to represent stimuli with higher probabilities, and for α > 0, neurons with higher firing rates are
used for stimuli with higher probabilities. Note also that the global resource values, N and R,
enter only as scale factors on the overall solution, allowing us to easily test the validity of the
5
(a)
Environment
(b)
Physiology
(c)
Threshold (deg)
# Cells
Probability
15
10
5
10
0
0
45
90
135
180
Perception
15
5
0
0
45
90
135
180
0
45
90
135
180
Orientation
(e) Discrimax predictions
Probability
(d) Infomax predictions
0
45
90
135
180
0
45
90
135
180
Orientation
Fig. 2. (a) Distribution of orientations averaged across three natural image databases [20–22]. (b)
Density, or total number of Macaque V1 cells tuned to each preferred orientation [23]. (c) Orientation
discrimination thresholds averaged across four human subjects [24]. (d & e) Infomax and discrimax
predictions of orientation distribution. Blue: prediction from cell density. Red: prediction from discrimination thresholds. Predictions were made by exponentiating the raw data with the appropriate
exponent from Table 1, then normalizing to integrate to one.
predicted relationships on experimental data. In addition to power-law relationships between tuning
properties and sensory priors, our formulation offers a direct relationship between the sensory prior
and perceptual discriminability. This can be obtained by substituting the optimal solutions for d(s)
and g(s) into Eq. (8),and using the resulting Fisher information to bound the discriminability,
δ(s) ≥ δmin (s) ≡ ∆/ If (s) [19]. The resulting expressions are provided in Table 1.
5
Experimental evidence
Our framework predicts a quantitative link between the sensory prior, physiological parameters (the
density, tuning widths, and gain of cells), and psychophysically measured discrimination thresholds.
We obtained subsets of these quantities for two visual stimulus variables, orientation and spatial
frequency, both of believed to be encoded by cells in primary visual cortex (area V1). For each
variable, we use the infomax and discrimax solutions to convert the physiological and perceptual
measurements, using the appropriate exponents from Table 1, into predictions of the stimulus prior
p̂(s). We then compare these predictions to the empirically measured prior p(s).
5.1
Orientation
We estimated the prior distribution of orientations in the environment by averaging orientation statistics across three natural image databases. Two databases consist entirely of natural scenes [20, 21],
and the third contains natural and manmade scenes [22]. Orientation statistics depend on scale, so
we measured statistics at a scale matching the psychophysical experiment from which we obtained
perceptual data. The average distribution of orientations exhibits higher probability at the cardinal
orientations (vertical and horizontal) than at the oblique orientations (Fig. 2(a)). Measurements of
cell density for a population of 79 orientation-tuned V1 cells in Macaque [23] show more cells tuned
to the cardinal orientations than the oblique orientations (Fig. 2(b)). Finally, perceptual discrimination thresholds, averaged across four human subjects [24] show a similar bias (Fig. 2(c)), with
humans better able to discriminate orientations near the cardinal directions.
All of the orientation data exhibit similar biases, but our theory makes precise and testable predictions about these relationships. If a neural population is designed to maximize information, then the
cell density and inverse discrimination thresholds should match the stimulus prior, as expressed in
infomax column of Table 1. We normalize these predictions to integrate to one (since the theory
6
Environment
(b)
Physiology
24
Log probability
100
tuning width
(cpd)
density
50 (# Cells)
12
0
0
5
10
15
Threshold (cpd)
(a)
0
0
1
2
3
4
5
6
(c)
Perception
0
5
3
2
1
0
7
10
15
Spatial Frequency (cpd)
(e) Discrimax predictions
Log probability
(d) Infomax predictions
0
5
10
15
0
5
10
15
Spatial Frequency (cpd)
Fig. 3. (a) Distribution of spatial frequencies computed across two natural image databases [20, 21].
(b) Cell density as a function of preferred spatial frequency for a population of 317 V1 cells [25,
28] Dark blue: average number of cells tuned to each spatial frequency. Light blue: average tuning
width. (c) Average spatial frequency discrimination thresholds. Dark red: thresholds obtained at
10% contrast averaged across 3 human subjects [26]. Light red: thresholds obtained at 25% contrast
averaged across 7-13 human subjects [27]. (d & e) Infomax and discrimax predictions of spatial
frequency distribution. Blues: predictions from cell density and tuning widths. Reds: predictions from
discrimination thresholds.
provides only the shapes of the functions, up to unknown values of the resource variables N and
R), and plot them against the measured prior (Fig. 2(d)). We see that the predictions arising from
cell density and discrimination thresholds are consistent with one another, and both are consistent
with the stimulus prior. This is especially remarkable given that the measurements come from very
different domains (in the case of the perceptual and physiological data, different species). For the
discrimax objective function, the exponents in the power-law relationships (expressed in Table 1)
are too small, resulting in poor qualitative agreement between the stimulus prior and predictions
from the physiology and perception (Fig. 2(e)). For example, predicting the prior from perceptual
data, under the discrimax objective function, requires exponentiating discrimination thresholds to
the fourth power, resulting in an over exaggeration of the cardinal bias.
5.2
Spatial frequency
We obtained a prior distribution over spatial frequencies averaged across two natural image
databases [20, 21]. For each image, we computed the magnitude spectrum, and averaged over orientation. We averaged these across images, and fit the result with a power law of exponent −1.3
(Fig. 3(a)). We also obtained spatial frequency tuning properties for a population of 317 V1 cells
[25]. On average, we see there are more cells, with correspondingly narrower tuning widths, tuned
to low spatial frequencies (Fig. 3(b)). These data support the model assumption that tuning width
is inversely proportional to cell density. We also obtained average discrimination thresholds for sinusoidal gratings of different spatial frequencies from two studies (Fig. 3(c)). The gratings were
shown at 10% contrast to 3 human subjects for one study [26], and 25% contrast for 7 − 13 human
subjects for the other [27]. The thresholds show that, on average, humans are better at discriminating
low spatial frequencies.
We again test the infomax and discrimax solutions by comparing predicted distributions obtained
from the physiological and perceptual data, to the measured prior. We normalize each prediction
to integrate to the corresponding area under the prior. The infomax case shows striking agreement
between the measured stimulus prior, and predictions based on the physiological and perceptual
measurements (Fig. 3(d)). However, as in the orientation case, discrimax predictions are poor (Fig.
3(e)), suggesting that information maximization provides a better optimality principle for explaining
the neural and perceptual encoding of spatial frequency than discrimination maximization.
7
6
Discussion
We have examined the influence sensory priors on the optimal allocation of neural resources, as
well as the influence of these optimized resources on subsequent perception. For a family of objective functions, we obtain closed-form solutions specifying power law relationships between the
probability distribution of a sensory variable encountered in the environment, the tuning properties
of a population that encodes that variable, and the minimum perceptual discrimination thresholds
achievable for that variable. We’ve shown preliminary supportive evidence for these relationships
for two different perceptual attributes.
Our analysis requires several approximations and assumptions in order to arrive at an analytical
solution. We first rely on lower bounds on mutual information and discriminability based on Fisher
information. Fisher information is known to provide a poor bound on mutual information when
there are a small number of neurons, a short decoding time, or non-smooth tuning curves [16, 29].
It also provides a poor bound on supra-threshold discriminability [30, 31]. But note that we do not
require the bounds on either information or discriminability to be tight, but rather that their optima
be close to that of their corresponding true objective functions. We also made several assumptions in
deriving our results: (1) the tuning curves, h(D(s)−n), evenly tile the stimulus space; (2) the single
neuron Fisher informations, φ(D(s) − n), evenly tile the stimulus space; and (3) the gain function,
g(s), varies slowly and smoothly over the width of φ(D(s) − n). These assumptions allow us to
approximate Fisher information in terms of cell density and gain (Fig. 1(e)), to express the resource
constraints in simple form, and to obtain a closed-form solution to the optimization problem.
Our framework offers an important generalization of the population coding literature, allowing for
non-uniformity of sensory priors, and corresponding heterogeneity in tuning and gain properties.
Nevertheless, it suffers from many of the same simplifications found in previous literature. First,
neural spike trains are not Poisson, and they are (at least in some cases) correlated [32]. Second,
tuning curve encoding models only specify neural responses to single stimulus values. The model
should be generalized to handle arbitrary combinations of stimuli. And third, the response model
should be generalized to handle multi-dimensional sensory inputs. Each of these limitations offers
an important opportunity for future work.
Finally, our encoding model has direct implications for Bayesian decoding, a problem that has received much attention in recent literature [e.g., 5, 6, 33–35]. A Bayesian decoder must have knowledge of prior probabilities, but it is unclear how such knowledge is obtained or represented in the
brain [34]. Previous studies assume that prior probabilities are either uniform [6], represented in
the spiking activity of a separate population of neurons [5], or represented (in sample form) in the
spontaneous activity [35]. Our encoding formulation provides a mechanism whereby the prior is implicitly encoded in the density and gains of tuning curves, which presumably arise from the strength
of synaptic connections. We are currently exploring the requirements for a decoder that can correctly
utilize this form of embedded prior information to obtain Bayesian estimates of stimulus variables.
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